MGMT 355, Spring 2012
Midterm 1 (Take home)
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1)
Name:
2)
Name:
3)
Name
Section sits at 11:30AM
1:00 PM
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All group members must sign stating that you ALL worked together in completing this
examination
(1)
In the matter of constructing control charts the following information is available
about some measurable quantity on a product (the units are omitted):
Sample # obs. 1
obs. 2
obs. 3
obs. 4
1
5
4
3
8
2
4
7
7
9
3
6
2
4
8
4
3
4
5
6
(a) Obtain the control limits for a three-sigma . (7.5 points)
(b) Similarly, obtain the control limits for a three-sigma R-chart. (7.5 points)
(c) Using a rough graph show how you would use these charts (you can draw one rough
graph for any one of these). (5 points)
(d) Suppose that you ask your machine operator to stop everytime the control limits of
your R-chart is violated. What fraction of the times would you find the machine to
have been stopped when actually nothing was wrong? (5 points)
(e) Contruct the control limits for a p-chart so that your charts have a failure-rate or falsealarm rate of 5%. Assume that using samples of size 64 you found to be 0.05. (10
points)
(f) Suppose you want to construct a control chart that will have 97.5% chance of
including sample means. What should be the UCL and LCL of the x-bar chart when
the average of the sample mean of the associated product is 36 oz and standard
deviation of sample means is 0.15 oz. (Use Excel to get an accurate value)
(2)
Joseph King has ambitions to be mayor of Williston, North Dakota. Joe has determined the
breakdown of the steps to the nomination and has estimated normal and crash costs and times
for the campaign as follows (times are in weeks).
Normal
A.
B.
C.
D.
E.
F.
G.
H.
Activity
Solicit Volunteers
Initial “Free” Exposure
Raise Money
Organize Schedule
Hire Advertising Firm
Arrange TV Interview
Advertising Campaign
Personal Campaigning
Crash
Time
6
3
9
4
2
3
5
7
Cost
$5,000
$4,000
$4,000
$1,000
$1,500
$4,000
$7,000
$8,000
Immediate Predecessors
Time
Cost
$9,000
4
--3
$4,000
--6
$10,000
A
2
$2,000
A
1
$2,000
B
1
$8,000
B
4
$12,000
C,E
5
$20,000
D,F
Joe King is not a wealthy man and would like to organize a 16-week campaign at minimum
cost. (i) Model his problem as a project network, (ii) identify the critical path employing
CPM. Using your solution of CPM, (iii) develop a crash plan to complete the campaign in the
desired time. Please note that you might have to draw multiple project network diagrams to
update your critical path with each crashing event by week. Your answer will show the EST,
EFT, LST and LFT (EST translates to « early start time » and it works the same for the other
acronyms).
(3)
(a)
The production process of a company has a process capability ratio () of 0.78.
What is the sigma rating of the quality of the production process? (5 points)
(b)
What is the quality rating in DPMO? (5 points)
(c)
The production cost of a good product is $28; bad products are resold to recover 25%
of cost. What is the effective average cost per unit of saleable product? What should be the
price earn 20% profit on effective cost? (10 points)
(d)
During the course of production, the maintenance engineer skips routine maintenance
on two consecutive dates. Investigation reveals the process capability index to have decreased
to 0.7, with more products falling below the LTL. What fraction of products is defective
under this condition of low maintenance? (Give a breakdown of defect types—below LTL
and above UTL. What is the effective cost per unit of saleable product? What is the loss per
unit incurred because of the process drift? (10 + 5=15 points)
(e)
Consider the situation when , when the process is centered. Suppose bad products
cannot always be detected perfectly; that there is only 85% chance that a bad product will be
discovered before sale. When a bad product is discovered, it is resold as before to recover
25% of cost. A bad product if sold to a customer has a 95% chance of being reported. When
reported, you replace the bad product free, refund 100% of the price and give a voucher
worth $20 as compensation for inconvenience. What is the effective cost per unit of revenue
generating product? What should be your price to earn a 20% profit on effective cost? (15
points)
(f) (This part has no contextual relation with the foregoing problems) A manufacturer
manufactures mechanical gears with the following radial specs : cm in radius. If its
production process is known to have a standard deviation of 0.005 cm, calculate its process
capability and its sigma-rating. (5+5=10 points)
(4)
Wegafolks Groceries usually has three cashiers working on Monday afternoons. The
customers arrive at the checkout area at a Poisson rate of fifteen every half an hour, and
randomly choose one of the lines dedicated to each of the cashiers. The cashiers can checkout
a customer on the average in five minutes; the service time follows an exponential probability
distribution.
(a)
your
How often is a cashier idle (any one)? (5 points) (Specify the model you use and show
calculation)
(b)
total
What is the average number of customers waiting for each of cashiers, and average
customers waiting in the check out area as a whole? (7.5 +2.5=10points)
(c)
What is the probability that an arriving customer will not have to wait to check out?
(Hint: probability that all cashiers are busy; note that they are independent variables)
(7.5 points)
(d)
What is the average time it takes a customer to check out? (5 points)
(8)
Wegfolks expects that the customer arrival rate will increase and will be 54 per hour
after the
opening of a new processed food section. In view of this it wants to investigate the
effect of
setting up a queue structure like airports---one queue serving all the cashiers. In doing
all the
following problems assume this increased customer arrival rate for all questions that
follow
(a)
What is the minimum number of cashiers it must hire to avoid overcrowding ,
that is
avoid forming infinitely long lines?
(5 points)
For the rest of the questions assume that it has now hired FIVE cashiers and there is
only
ONE queue in the store.
(a) What would be the average waiting time for a customer? (10 points)
(b) What fraction of customers would have to get in the queue before check out? (10
points)
(c) On the average how much time must a customer budget to check out? (5 points)
(5) Consider the followind data about a simple project (note that the table gives the expected
time and the standard deviation of completion time for each activity). (a) Draw the
corresponding project network AS WE HAVE DONE IN CLASS. (b) Identify the critical
path, (c) specify expected project completion time and (d) the standard deviation of the
project completion time. (e) What is the chance that the project would be completed between
40 and 45 weeks to complete? (f) Imagine that you are a contractor and plan to place a bid for
a project contract. (g)Apart from a price offer you need to propose a project completion time
that you want to achieve with 99% certainty, what would be the minimum time you will state
in you bid ?(5+10 (b,c)+5+5+5+10)
Activity
A
B
C
D
E
F
G
H
Immediate
Predecessor(s)
----A
B
B
C,D
D,E
G
Expected
Time(Weeks)
4
5
7
10
7
8
9
10
Standard Dev. of
Time (Weeks)
3
1
3
2
1.5
2.2
4
2
(6) Mrs. Rich has just bought a new $30,000 car. As a reasonably safe driver, she believes
that there is only a 5% chance of being in an accident in the forthcoming year. If she is
involved in an accident, the damage to her new car depends on the severity of the accident.
The probability distribution for the range of possible accidents and the corresponding damage
amounts (in dollars) are shown in the table below. Mrs. Rich is trying to decide whether she
is willing to pay $170 each year for collision insurance with a $300 deductible. Note that with
this type of insurance, she pays the first $300 in damages if she causes an accident, and the
insurance company pays the remainder.
Distribution of Accident Types and Corresponding Damage Amounts
Type of Accident
Minor
Moderate
Serious
Catastrophic
Conditional Probability
0.60
0.20
0.10
0.10
Damage to Car
$200
$1,000
$4,000
$30,000
(i) Model her decsion problem as a decision tree. (8)
(ii) Identify her optimal decision for the given deductible.(12)
(iii)
Find the maximum deductible that will make her indifferent between buying the
policy and not buying the policy. (5)
(7) Consider an oil-wildcatting problem. You have mineral rights on a piece of land that you
believe may have oil underground. There is only a 10% chance that you will strike oil if
you drill, but the the payoff is $200,000. It costs $10,000 to drill. The alternative is not to
drill at all, in which case your profit is zero.
(a) Draw a decision tree to model your decision problem and determine your optimal
decision
and the value of the decsion problem. (5)
(b) Now suppose you have access to the services of the « omniscient Yoda » who blinks
his
right eye if he sees oil in your land and blinks his left eye if he doesn’t see oil in your
land.
In short, Yoda guve you perfect information before you can make your decision. What
is the
value of perfect information provided by Yoda. What use is this information to you ?
(10)
(c) Now suppose that before you drill, you might consult a geologist who can assess the
promise
of the piece of land. She can tell you whether your prospects are « good » or « poor ».
But,
unlike Yoda, she is not a perfect predictor. If there is oil there is a 95% chance that
she will
say the prospects are « good » and if there is no oil then there is only a 10% chance
that she
will say that the prospects are « good ».
(i)
consult
Draw an augmented decision tree that also included the decision of whether to
the geologist or not. (10)
(i)
What is your optimal decision (consulting or not consulting the geologist) and
the
associated action strategy ? (20)
(iii)
What is the EVSI and the efficiency of the information ? (5+5=10)
(iv)
If she charges $7000, would you hire her ? (5)
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